# Filed Under #cooperation

Nov 29, 2022

### Example numerical analysis of replicator dynamics with more than 2 strategies

Recently, I set a task for a student to use Python to analyse the replicator dynamics of a game with three strategies, including using the Jacobian to determine the stability of the steady state. The purpose of this blog post is to share the solution in case that’s useful to someone. The game For this task, we will consider the replicator dynamics of a 2-player game with 3 strategies. Recall the general equation for the replicator dynamics is \[\begin{equation} \dot{p}_i = p_i (f_i - \bar{f}) \end{equation}\] where \(p_i\) is proportion of \(i\)-strategists in the population, \(f_i\) is fitness effect of...Nov 10, 2022

### Group nepotism and the Brothers Karamazov Game

I recently read a classic paper by anthropologist Jones (2000), ‘Group nepotism and human kinship’, which argues that collective action may explain some of the key features of human cooperation between kin. Collective action here is action by a group (e.g., clan, tribe) that has sufficient ingroup solidarity to decide how to act as a collective. The key features it may help explain are: 1. the ‘axiom of amity’, that one is obliged to help kin just because they’re kin; 2. that kin groups come in many sizes and can be quite large; and 3. that human notions of kinship...Aug 1, 2022

### Lab meeting about threshold games

Last month, I chose a paper by Kris De Jaegher in Scientific Reports for our weekly lab-meeting discussion. We used the paper to teach ourselves about threshold games, and the purpose of this blog post is to summarise the things we learnt. Replicator dynamics revision De Jaegher used the replicator dynamics approach. Replicator dynamics assumes a well-mixed, infinitely large population of players that reproduces asexually, where the number of offspring they produce depends upon the payoff they receive from a game-theoretic ‘game’, as illustrated below:– Replicator dynamics animation, created by HowieKor (Creative Commons) We start with a population of \(n\)...Apr 1, 2022

### Human cooperation and social network expansion over time

It is generally held that human cooperation first evolved in our ancestral past, when we tended to live in small groups composed mostly of family members, and when cooperation could therefore be selected for by kin selection. This narrative is also sometimes invoked to explain the origin of cooperative mechanisms that are primarily about cooperation between nonkin. For example, in the iterated Prisoner’s Dilemma, cooperation between nonkin can be maintained by the tit-for-tat strategy, and a population of tit-for-tat players can resist invasion by defectors. However, tit-for-tat strategists cannot invade a population of defectors (presumably the primordial strategy), and so...Mar 1, 2022

### Stochastic evolutionary dynamics of the Volunteers' Dilemma

The purpose of this blog post is to use a recent paper by Tutić (2021) to teach myself about stochastic evolutionary game theory. Tutić’s (2021) model concerns the Volunteer’s Dilemma, which is a public goods game where the public good is provided if at least one group member cooperates. In the replicator dynamics, cooperators can always invade a population of defectors. However, as the group size increases, the proportion of cooperators in the population at the evolutionary steady-state declines (see Tutić (2021) Fig. 6), which increases the risk that cooperators will be lost from a finite population where stochastic forces...Feb 1, 2022

### Using coalescence models to approximate interaction probabilities in weak selection

To calculate the conditions under which selection favours cooperation, Antal et al. (2009) used interaction probabilities calculated from a coalescence model. However, a coalescence model is a neutral model, it assumes no selective differences between types. So why can they assume no selective difference between types in a selective model? The purpose of this blog post is to explore the explanation detailed in their SI. First, a quick overview of the paper. The model involves the simultaneous evolution of two traits: (1) the strategy in a one-shot Prisoner’s Dilemma, either Cooperate or Defect; and (2) a phenotypic tag, modelled as...Dec 12, 2021

### Connection between replicator dynamics, Bernstein polynomials, and Bezier curves

A series of papers by Jorge Peña and others provide a neat way to characterise the replicator dynamics of two-strategy nonlinear public goods games with random group formation (Peña et al. 2014, Peña et al. 2015, Peña & Nöldeke 2016, Nöldeke & Peña 2016, Peña & Nöldeke 2018, Nöldeke & Peña 2020). I this blog post, I used a paper by Archetti (2018) to teach myself more about the topic. Archetti (2018) considered replicator dynamics in a population of producers (cooperators) and defectors. The probability that \(i\) of the \(n-1\) other members in your group are cooperators is \[p_{i,n-1}(x) =...Nov 18, 2021

### Cooperation and maladaptation

Humans often cooperate in experimental games even when it’s not the strategy that will win them the most money (Ledyard, 1995; Cadsby and Maynes, 1999). They do this in one-shot games, where there is no chance the other could punish them if they took advantage, and in games with anonymous strangers, for whom they have no material reason to care. We see this cooperative tendency typically in the initial rounds of experimental public goods games (e.g., Offerman et al. 1996), but also in real-world common-resource problems (Ostrom 2000). From a naive economic or Darwinian perspective, which expects humans to behave...Aug 23, 2021

### Combining a stake in the future with the presence of others negatively affects intergenerational cooperation

Conservationists often urge us to think about what kind of world we want to leave to future generations. But what if thinking about future generations — and about our descendants, specifically — triggers a geneological self-interest in us that can cause the environmental message to backfire? That was one of the questions raised by Chia-chen Chang’s new paper in Royal Society Open Science. The paper concerns results from a lab-based intergenerational common-pool resource game (Fischer et al. 2004, Hauser et al. 2014, Lohse et al. 2020). A common-pool resource game is a game that is designed to mimic the behaviour...May 24, 2021

### Intergenerational reciprocity and restoration

In conservation, we often talk about intergenerational equity, which is the idea that we should behave sustainably out of fairness towards future generations. However, how we treat others often depends on how we have been treated ourselves. If previous generations have over-exploited the natural resources and behaved unsustainably, will that make us feel less inclined to behave sustainably for future generations? In Chia-chen Chang’s recent paper in Sustainability Science, we explored this question using an experimental common-pool resource game. An example of a common-pool resource game The paper is specifically interested in restoration, and the key innovation was to investigate...Apr 1, 2021

### Homophily

Homophily is the principle that contact between similar people occurs at a higher rate than among dissimilar people (reviewed in McPherson et al., 2001). It is a long-established principle (see e.g., Byrne (1969), which has 1000+ citations), and is particularly well established for friendships, which have been shown to be homophilous for ethnicity, age, religion, education, and social values ([16,156–160] in Dunbar (2018)). ‘Baseline’ homophily occurs because of the demography of the potential relationship pool, and effects beyond that occur due to other factors not limited to personal preference. Homophily acts on a variety of different dimensions. The ‘Big 5’...Feb 1, 2021

### Evolutionary game theory model with homophilic imitation

The purpose of this blog post is to replicate results from Vasconcelos et al. (2014) and use it to learn about the behaviour of this kind of model: Vasconcelos, V. V., Santos, F. C., Pacheco, J. M. and Levin, S. A. (2014). Climate policies under wealth inequality, Proceedings of the National Academy of Sciences 111(6): 2212–2216. I am interested in this model because it involves homophily: it is a social learning model, but the probability of imitation depends not only on the relative payoffs between the focal player and other but also on how similar they are in wealth. Code...Oct 7, 2020

### Using n-player relatedness to model the Modern Tragedy of the Commons

In this blog post, I will use the “Modern Tragedy of the Commons” as an example for how to use Hisashi Ohtsuki’s \(n\)-player generalisation of relatedness. In a recent experimental study in Science Advances, Gross and De Dreu (2019) described the Modern Tragedy of the Commons: if people can individually solve a problem by purchasing a solution that protects themselves only, then they may prefer that to contributing to a cheaper collective solution (Fig. 1). One recent example might be the trend among wealthy Californian home-owners to hire private firefighters to defend their property from wildfires exacerbated by climate change....Aug 16, 2018

### Fixation probability of birth-death process

The goal is to understand where Eq. 2 of the Supplementary section of Sigmund et al. (2010) came from. We are considering a finite population within which individuals are pursuing different game-theoretic strategies. At each timestep, a pair of individuals is chosen at random, and they engage in a social learning process, where individual \(i\) will adopt the strategy of individual \(j\) according to a function that increases with difference in the payoff betweeen the two strategies \(P_j - P_i\), \[\frac{1}{1+\exp( -s (P_j-P_i) )}\] where \(0 \leq s < \infty\) is the imitation strength parameter. Let’s say that we have...Dec 15, 2016

### Understanding the neigbour-modulated inclusive fitness approach

The goal is to understand Equation 4.2 of Rodrigues and Kokko (2016). To understand the technique, I read through Taylor et al. (2007), and Taylor and Frank (1996), specifically examples 4, 4a and 4b. Very briefly, we begin with a matrix \(A = [w_{i,j} ]\) whose elements represent the genetic contribution of class \(j\) to class \(i\). Then for some mutant genic value \(x\), the fitness derivative is \[\frac{dW}{dx} = \sum_{i,j} v_i \frac{dw_{i,j}}{dx} u_j\] where \(\mathbf{v}\) is the left eigenvector of \(A\) and is the reproductive values, and \(\mathbf{u}\) is the right eigenvector of \(A\) and is the class frequencies....Aug 23, 2016

### Evolution of parochialism

Previously I was reading about identity in Singapore, and I found that the government project of creating a national identity is focused upon creating a cosmopolitan mindset (i.e. one of openness and tolerance of difference), and that this interacts in different ways with the two main groups of foreigners, the ‘foreign talent’ and the ‘foreign worker’. The literature review that I did there was focused upon the sociological literature and Singapore specifically. In this post I provide a summary of some reading that I have done on experiments investigating group identity using economic games. I’ve focused specifically upon parochialism in...Jul 23, 2016

### Identity in Singapore, the Internet, and connections to game theory

The following is a literature summary that I undertook as part of my broader interest in modelling social systems, using identity in Singapore as a case study. I should preface all of this by saying that the ideas below are not ‘my own’ as such, but rather my attempt to synthesise the literature and opinions of experts in Singapore. Identity in Singapore Since independence, the Singaporean government has actively worked to construct a civic national identity for the country. Three main phases can be identified (Ortmann, 2009): (1) 1965-1980s, focused primarily upon the objective of economic growth, the idea of...Mar 23, 2016